The representations (2) and (3) may be interpreted as such polynomials as intended.

Multiplying the equations (2) and (3) we obtain a corresponding representation for the square root of m⁢nmnmn which also lies in the quartic fieldℚ⁢(m,n)=ℚ⁢(m+n)ℚmnℚmn\mathbb{Q}(\sqrt{m},\,\sqrt{n})=\mathbb{Q}(\sqrt{m}\!+\!\sqrt{n}):

Remark. The sum (1) of two square roots of positive squarefree integers is always irrational, since in the contrary case, the equation (3) would say that nn\sqrt{n} would be rational; this has been proven impossible here.

Related:

BinomialTheorem

Synonym:

expressing two square roots with their sum, irrational sum of square roots